A summer resort rents rowboats to customers but does not allow more than four pe

Question

A summer resort rents rowboats to customers but does not allow more than four people to a boat. Each boat is designed to hold no more than 800 pounds. Suppose the distribution of adult males who rent boats, including their clothes and gear, is normally distributedwith a mean of 190 pounds and standard deviation of 10 pounds.

1) How do we know that the sample mean of four adult male passengers is normally distributed in thisproblem?

2)What is the probability that a group of four adult male passengers will exceed the weight limit of 800 pounds?

3) The resort is concerned that their boats’ 800-lbs capacity may not be good enough and they plan to order new boats with a higher weight limit. They want to make sure that a group of four adult males will not exceed the weight limit of the new boats 99.9% of the time. What weight capacity should the new boats have?

Please include as detailed as possible answers :) My class only uses graphing calculator technology-- not z score tables or excel

Explanation / Answer

When you add independent normally distributed random variables, the mean of their sum is the sum of their means and the variance of their sum is the sum of their variances.

Thus, for the group of four adult male passengers, their total weight has a mean of 190 x 4 = 760 pounds, and variance of (10^2 x 4) = 400 pounds^2. The standard deviation is the square root of the variance, so it is 400^(1/2) = 20 pounds.

Now use your normal tables or calculator to find
P(total weight > 800 pounds).

If you use standardized normal tables, then

z = (x - mean) / (standard deviation)
= (800 - 760) / 20
= 40 / 20
= 2.

This is a familiar number, and the probability of exceeding 2 standard deviations above the mean is 0.02275 (to five decimal places).

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